arXiv subjects

Shixu Meng

Publications and source records attributed to Shixu Meng.

At least 19 recordsLinked to original sources

arXiv2026

Characterizing nonlinear information in the linear sampling method for inverse medium scattering

This work is concerned with the nonlinear information in the linear sampling method for the inverse medium scattering problem. In addition to the well-known capability in shape characterization, we demonstrate that the imaging indicator represents nonlinear information about the unknown contrast. We further demonstrate additional representations for this nonlinear information using the generalized linear sampling method. We shed light on this new result using the Born model and analytical examples.

math.AP↗

arXiv2026

Inverse scattering beyond Born approximation via rotation-equivariance-aware neural network and low-rank structure

This work proposes a hybrid method (ULR) which integrates a rotation-equivariance-aware neural network and a low-rank structure to solve the two dimensional inverse medium scattering problem. The neural network is to model the data corrector which maps the full data to the Born data, and the low-rank structure is to design an inverse Born solver that finds a regularized solution from the perturbed Born data. The proposed rotation-equivariance-aware neural network naturally incorporates the reciprocity relation and the rotation-equivariance in inverse scattering, while the low-rank structure effectively filters high-frequency noise in the output of the neural network and leads to a regularized method supported by theoretical stability in the Born region. For a comparative study, we replace the low-rank inverse Born solver by another rotation-equvariance-aware neural network to propose a two-step neural network (UU). Furthermore, we extend the proposed methods (ULR and UU) to tackle the more challenging case with only limited aperture data. A variety of numerical experiments are conducted to compare the proposed ULR, UU, and a black-box neural network.

math.NA↗

arXiv2026

Low-rank-assisted inverse medium scattering: Lipschiz stability and ensemble Kalman filter

In this work we study the theoretical Lipschitz stability and propose a low-rank-assisted numerical method for the inverse medium scattering beyond the Born region. The proposed low-rank structure is based on the disk prolate spheroidal wave functions, which are eigenfunctions of both the Born forward operator and a Sturm-Liouville differential operator. We obtain Lipschitz stability for unknowns in a low-rank space in the fully nonlinear case and characterize the explicit Lipschitz constant in the linearized region. We further propose an ensemble Kalman filter to iteratively update the unknown in the proposed low-rank space whose dimension is intrinsically determined by the wave number. Moreover the ensembles are sampled according to a novel trace class covariance operator motivated by the connection between the proposed low-rank space and the Sturm-Liouville differential operator. Finally numerical examples are provided to illustrate the feasibility of the proposed method.

math.NA↗

arXiv2026

A theoretical and computational framework for three dimensional inverse medium scattering using the linearized low-rank structure

In this work we propose a theoretical and computational framework for solving the three dimensional inverse medium scattering problem, based on a set of data-driven basis arising from the linearized problem. This set of data-driven basis consists of generalizations of prolate spheroidal wave functions to three dimensions (3D PSWFs), the main ingredients to explore a low-rank approximation of the inverse solution. We first establish the fundamentals of the inverse scattering analysis, including regularity in a customized Sobolev space and new a priori estimate. This is followed by a computational framework showcasing computing the 3D PSWFs and the low-rank approximation of the inverse solution. These results rely heavily on the fact that the 3D PSWFs are eigenfunctions of both a restricted Fourier integral operator and a Sturm-Liouville differential operator. Furthermore we propose a Tikhonov regularization method with a customized penalty norm and a localized imaging technique to image a targeting object despite the possible presence of its surroundings. Finally various numerical examples are provided to demonstrate the potential of the proposed method.

math.NA↗

arXiv2025

On the surface Bloch waves in truncated periodic media: scalar-wave primer

Much like their counterparts in homogenous elastic solids, waves in periodic media can be broadly classified into Floquet-Bloch body waves, and evanescent surface waves. Our goal is to elucidate the latter boundary layers, termed surface Bloch (SB) waves, affiliated with rational surface cuts and homogeneous Neumann data. To this end we adopt a two-dimensional (2D) scalar wave equation with periodic coefficients (describing anticline shear waves in phonoic crystals) as a test bed and develop a unit cell-of-periodicity-based, reduced order model of the SB waves that is capable of describing both their dispersion, waveforms, and ``skin depth''. The centerpiece of our analysis is a quadratic eigenvalue problem (QEP) for the effective unit cell of periodicity -- deriving from a geometric interplay between the mother Bravais lattice and orientation of the surface cut -- that seeks the complex wavenumber normal to the cut plane given (i) the excitation frequency and (ii) wavenumber in the direction of the cut plane. In this way the sought boundary layer is derived via superposition of the evanescent QEP eigenstates, whose relative amplitudes are obtained by imposing the homogeneous boundary condition. With the QEP eigenspectrum at hand, evaluation of an SB wave -- in terms of both dispersion characteristics and evanescent waveforms -- entails only a low-dimensional eigenvalue problem. This feature caters for rapid exploration of the effect of (periodic) surface undulations, and so enables manipulation of the SB waves via optimal design of the surface cut. Our analysis also includes an account for the power flow and ``skin depth'' of a surface Bloch wave, both of which are critical for the energetic relevance of boundary layers.

physics.app-ph↗

arXiv2025

On the recovery of two function-valued coefficients in the Helmholtz equation for inverse scattering problems via inverse Born series

In this work, we construct the Born and inverse Born approximation and series to recover two function-valued coefficients in the Helmholtz equation for inverse scattering problems from the scattering data at two different frequencies. An analysis of the convergence and approximation error of the proposed regularized inverse Born series is provided. The results show that the proposed series converges when the inverse Born approximations of the perturbations are sufficiently small. The preliminary numerical results show the capability of the proposed regularized inverse Born approximation and series for recovering the isotropic inhomogeneous media.

math.NA↗

arXiv2024

An inverse obstacle scattering problem with passive data in the time domain

This work considers a time domain inverse acoustic obstacle scattering problem due to passive data. Motivated by the Helmholtz-Kirchhoff identity in the frequency domain, we propose to relate the time domain measurement data in passive imaging to an approximate data set given by the subtraction of two scattered wave fields. We propose a time domain linear sampling method for the approximate data set and show how to tackle the measurement data in passive imaging. An imaging functional is built based on the linear sampling method, which reconstructs the support of the unknown scattering object using directly the time domain measurements. The functional framework is based on the Laplace transform, which relates the mapping properties of Laplace domain factorized operators to their counterparts in the time domain. Numerical examples are provided to illustrate the capability of the proposed method.

math.NA↗

arXiv2024

Exploring low-rank structure for an inverse scattering problem with far-field data

In this work, we introduce a novel low-rank structure tailored for solving the inverse scattering problem. The particular low-rank structure is given by the generalized prolate spheroidal wave functions, computed stably and accurately via a Sturm-Liouville problem. We first process the far-field data to obtain a post-processed data set within a disk domain. Subsequently, the post-processed data are projected onto a low-rank space given by the low-rank structure. The unknown is approximately solved in this low-rank space, by dropping higher-order terms. The low-rank structure leads to an explicit stability estimate for unknown functions belonging to standard Sobolev spaces, and a Lipschitz stability estimate for unknowns belonging to a finite dimensional low-rank space. Various numerical experiments are conducted to validate its performance, encompassing assessments of resolution capability, robustness against randomly added noise and modeling errors, and demonstration of increasing stability.

math.NA↗

arXiv2024

Preconditioning Low Rank Generalized Minimal Residual Method (GMRES) for Implicit Discretizations of Matrix Differential Equations

This work proposes a new class of preconditioners for the low rank Generalized Minimal Residual Method (GMRES) for multiterm matrix equations arising from implicit timestepping of linear matrix differential equations. We are interested in computing low rank solutions to matrix equations, e.g. arising from spatial discretization of stiff partial differential equations (PDEs). The low rank GMRES method is a particular class of Krylov subspace method where the iteration is performed on the low rank factors of the solution. Such methods can exploit the low rank property of the solution to save on computational and storage cost. Of critical importance for the efficiency and applicability of the low rank GMRES method is the availability of an effective low rank preconditioner that operates directly on the low rank factors of the solution and that can limit the iteration count and the maximal Krylov rank. The preconditioner we propose here is based on the basis update and Galerkin (BUG) method, resulting from the dynamic low rank approximation. It is a nonlinear preconditioner for the low rank GMRES scheme that naturally operates on the low rank factors. Extensive numerical tests show that this new preconditioner is highly efficient in limiting iteration count and maximal Krylov rank. We show that the preconditioner performs well for general diffusion equations including highly challenging problems, e.g. high contrast, anisotropic equations. Further, it compares favorably with the state of the art exponential sum preconditioner. We also propose a hybrid BUG - exponential sum preconditioner based on alternating between the two preconditioners.

math.NA↗

arXiv2024

Plug-and-play analytical paradigm for the scattering of plane waves by "layer-cake" periodic systems

We investigate the scattering of scalar plane waves in two dimensions by a heterogeneous layer that is periodic in the direction parallel to its boundary. On describing the layer as a union of periodic laminae, we develop a solution of the scattering problem by merging the concept of propagator matrices and that of Bloch eigenstates featured by the unit cell of each lamina. The featured Bloch eigenstates are obtained by solving the quadratic eigenvalue problem (QEVP) that seeks a complex-valued wavenumber normal to the layer boundary given (i) the excitation frequency, and (ii) real-valued wavenumber parallel to the boundary -- that is preserved throughout the system. Spectral analysis of the QEVP reveals sufficient conditions for discreteness of the eigenvalue spectrum and the fact that all eigenvalues come in complex-conjugate pairs. By deploying the factorization afforded by the propagator matrix approach, we demonstrate that the contribution of individual eigenvalues (and so eigenmodes) to the solution diminishes exponentially with absolute value of their imaginary part, which then forms a rational basis for truncation of the factorized Bloch-wave solution. The proposed methodology caters for the optimal design of rainbow traps, energy harvesters, and metasurfaces, whose potency to manipulate waves is decided not only by the individual dispersion characteristics of the component laminae, but also by ordering and generally fitting of the latter into a composite layer. Using the factorized Bloch approach, evaluation of trial configurations -- as generated by the permutation and window translation/stretching of the component laminae -- can be accelerated by decades.

math-ph↗

arXiv2024

Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator

In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.

math.NA↗

arXiv2023

Data-driven basis for reconstructing the contrast in inverse scattering: Picard criterion, regularity, regularization, and stability

We consider the inverse medium scattering of reconstructing the medium contrast using Born data, including the full aperture, limited-aperture, and multi-frequency data. We propose data-driven basis functions for these inverse problems based on the generalized prolate spheroidal wave functions and related eigenfunctions. Such data-driven eigenfunctions are eigenfunctions of a Fourier integral operator; they remarkably extend analytically to the whole space, are doubly orthogonal, and are complete in the class of band-limited functions. We first establish a Picard criterion for reconstructing the contrast using the data-driven basis, where the reconstruction formula can also be understood from the viewpoint of data processing and analytic extrapolation. Another salient feature associated with the generalized prolate spheroidal wave functions is that the data-driven basis for a disk is also a basis for a Sturm-Liouville differential operator. With the help of Sturm-Liouville theory, we estimate the $L^2$ approximation error for a spectral cutoff approximation of $H^s$ functions. This yields a spectral cutoff regularization strategy for noisy data and an explicit stability estimate for contrast in $H^s$ ($0<s<1/2$) in the full aperture case. In the limited-aperture and multi-frequency cases, we also obtain spectral cutoff regularization strategies for noisy data and stability estimates for a class of contrast.

math.NA↗

arXiv2023

Single Mode Multi-frequency Factorization Method for the Inverse Source Problem in Acoustic Waveguides

This paper investigates the inverse source problem with a single propagating mode at multiple frequencies in an acoustic waveguide. The goal is to provide both theoretical justifications and efficient algorithms for imaging extended sources using the sampling methods. In contrast to the existing far/near field operator based on the integral over the space variable in the sampling methods, a multi-frequency far-field operator is introduced based on the integral over the frequency variable. This far-field operator is defined in a way to incorporate the possibly non-linear dispersion relation, a unique feature in waveguides. The factorization method is deployed to establish a rigorous characterization of the range support which is the support of source in the direction of wave propagation. A related factorization-based sampling method is also discussed. These sampling methods are shown to be capable of imaging the range support of the source. Numerical examples are provided to illustrate the performance of the sampling methods, including an example to image a complete sound-soft block.

math.NA↗

arXiv2021

A multi-frequency sampling method for the inverse source problems with sparse measurements

We consider the inverse source problems with multi-frequency sparse near field measurements. In contrast to the existing near field operator based on the integral over the space variable, a multi-frequency near field operator is introduced based on the integral over the frequency variable. A factorization of this multi-frequency near field operator is further given and analysed. Motivated by such a factorization, we introduce a multi-frequency sampling method to reconstruct the source support. Its theoretical foundation is then derived from the properties of the factorized operators and a properly chosen point spread function. Numerical examples are provided to illustrate the multi-frequency sampling method with sparse near field measurements. Finally we briefly discuss how to extend the near field case to the far field case.

math.NA↗

arXiv2021

Modified sampling method with near field measurements

This paper investigates the inverse scattering problems using sampling methods with near field measurements. The near field measurements appear in two classical inverse scattering problems: the inverse scattering for obstacles and the interior inverse scattering for cavities. We propose modified sampling methods to treat these two classical problems using near field measurements without making any asymptotic assumptions on the distance between the measurement surface and the scatterers. We provide theoretical justifications based on the factorization of the near field operator in both symmetric factorization case and non-symmetric factorization case. Furthermore, we introduce a data completion algorithm which allows us to apply the modified sampling methods to treat the limited-aperture inverse scattering problems. Finally numerical examples are provided to illustrate the modified sampling methods with both full- and limited- aperture near field measurements.

math.NA↗

arXiv2021

Data completion algorithms and their applications in inverse acoustic scattering with limited-aperture backscattering data

We introduce two data completion algorithms for the limited-aperture problems in inverse acoustic scattering. Both completion algorithms are independent of the topological and physical properties of the unknown scatterers. The main idea is to relate the limited-aperture data to the full-aperture data via the prolate matrix. The data completion algorithms are simple and fast since only the approximate inversion of the prolate matrix is involved. We then combine the data completion algorithms with imaging methods such as factorization method and direct sampling method for the object reconstructions. A variety of numerical examples are presented to illustrate the effectiveness and robustness of the proposed algorithms.

math.AP↗

arXiv2021

The Spectral Analysis of the Interior Transmission Eigenvalue Problem for Maxwell's Equations

In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. We study this problem in the framework of semiclassical analysis and relate the transmission eigenvalues to the spectrum of a Hilbert-Schmidt operator. Under the additional assumption that the contrast is constant in a neighborhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space.

math.AP↗

arXiv2021

A convergent low-wavenumber, high-frequency homogenization of the wave equation in periodic media with a source term

We pursue a low-wavenumber, second-order homogenized solution of the time-harmonic wave equation at both low and high frequency in periodic media with a source term whose frequency resides inside a band gap. Considering the wave motion in an unbounded medium $\mathbb{R}^d$ ($d\geqslant1$), we first use the (Floquet-)Bloch transform to formulate an equivalent variational problem in a bounded domain. By investigating the source term's projection onto certain periodic functions, the second-order model can then be derived via asymptotic expansion of the Bloch eigenfunction and the germane dispersion relationship. We establish the convergence of the second-order homogenized solution, and we include numerical examples to illustrate the convergence result.

math.AP↗